Salut Pouvez vous m′aider avec ce devoir? Examiner si les series suivantes sont absolument convergentes ou semi−convergentes Σ_(k=0) ^n (((-1)^(k-1) )/((k+1)^2 )) Σ_(k=0) ^n (((-1)^(k-1) )/k) Etudier la convergence des series suivantes Σ_(k=1) ^n (((2k)!)/2^k ) Σ


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Question Number 118952 by Backer last updated on 21/Oct/20

$Salut$ $Pouvez vous m^{'} aider avec ce devoir ?$ $Examiner si les series suivantes sont$ $absolument convergentes ou$ $semi - convergentes$ $\underset{k = 0}{\sum^{n}} \frac{{(- 1)}^{k - 1}}{{(k + 1)}^{2}}$ $\underset{k = 0}{\sum^{n}} \frac{{(- 1)}^{k - 1}}{k}$ $Etudier la convergence des series suivantes$ $\underset{k = 1}{\sum^{n}} \frac{(2 k)!}{2^{k}}$ $\underset{k = 1}{\sum^{n}} {(\frac{1}{2})}^{k}$
	Answered by Bird last updated on 21/Oct/20

	$\sum_{k = 0}^{\infty} \frac{{(- 1)}^{k - 1}}{{(k + 1)}^{2}} e s t a b s o l u m e n t$ $c o n v e r g e n t c a r ∣ \frac{{(- 1)}^{k - 1}}{k + 1} ∣⩽ \frac{1}{{(k + 1)}^{2}}$ $Σ \frac{{(- 1)}^{k - 1}}{k} i s s l t e r n a t e s e r i e$ $c o n v e r g e n t a t f o r m Σ {(- 1)}^{k - 1} v_{k}$ $v_{k} ⩾ 0 a n d v_{k} d e c r e a z e t o 0$
	Answered by Bird last updated on 21/Oct/20

	$n a t u r e d e l a s e r i e \sum_{n = 1}^{\infty} \frac{(2 n)!}{2^{n}}$ $\frac{u_{n + 1}}{u_{n}} = \frac{(2 n + 2)!}{2^{n + 1}} \times \frac{2^{n}}{(2 n)!}$ $= \frac{(2 n + 2) (2 n + 1)}{2} \sim 2 n^{2} \to + \infty$ $\Rightarrow c e t t e s e r i e i s d i v e r g e n t e$
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